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The transformed $l_1$ penalty (TL1) functions are a one parameter family ofbilinear transformations composed with the absolute value function. When actingon vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm($p \in (0,1)$). In our companion paper, we showed that TL1 is a robustsparsity promoting penalty in compressed sensing (CS) problems for a broadrange of incoherent and coherent sensing matrices. Here we develop an explicitfixed point representation for the TL1 regularized minimization problem. TheTL1 thresholding functions are in closed form for all parameter values. Incontrast, the $l_p$ thresholding functions ($p \in [0,1]$) are in closed formonly for $p=0,1,1/2,2/3$, known as hard, soft, half, and 2/3 thresholdingrespectively. The TL1 threshold values differ in subcritical (supercritical)parameter regime where the TL1 threshold functions are continuous(discontinuous) similar to soft-thresholding (half-thresholding) functions. Wepropose TL1 iterative thresholding algorithms and compare them with hard andhalf thresholding algorithms in CS test problems. For both incoherent andcoherent sensing matrices, a proposed TL1 iterative thresholding algorithm withadaptive subcritical and supercritical thresholds consistently performs thebest in sparse signal recovery with and without measurement noise.

Minimization of Transformed $L_1$ Penalty: Closed Form Representation and Iterative Thresholding Algorithms